The adjoint action is observable

Let G be a connected algebraic group and consider the adjoint action μ:G×G→G defined by μ(g,x)=gxg−1. We show that this action is observable. A major step in our proof is to show that the adjoint action is induced generically from the conjugating action of NG(T) (the normalizer of a maximal torus) on a certain open subset of CG(T) (the centralizer of T). As a direct consequence we observe that the ring of invariants k[G]G⊂k[G] separates the conjugacy classes of G generically.